Orateur
Description
One of the many connections between Grassmannians and combinatorics is cohomological: The cohomology ring of a Grassmannian ${\rm Gr}(k,n)$ is a quotient of the ring $S$ of symmetric polynomials in $k$ variables. More precisely, it is the quotient of $S$ by the ideal generated by the k consecutive complete homogeneous symmetric polynomials $h_{n-k}, h_{n-k+1}, \ldots , h_n$. We deform this quotient, by replacing the ideal by the ideal generated by $h_{n-k} - a_1 , h_{n-k+1} - a_2 , \ldots , h_n - a_k$ for some $k$ fixed elements $a_1 , a_2 , \ldots , a_k$ of the base ring. This generalizes both the classical and the quantum cohomology rings of ${\rm Gr}(k,n)$. We find three bases for the new quotient, as well as an $S_3$-symmetry of its structure constants, a “rim hook rule” for straightening arbitrary Schur polynomials, and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the $a_i$ as signed indeterminate), which suggests a geometric or
combinatorial meaning for the quotient.
